Frequency references provided by oscillators are required in every clocked electronic system, including communication circuits, microprocessors, and signal processing circuits. Oscillators frequently consist of high performance piezoelectric crystals, such as quartz oscillators. The advantages of quartz oscillators are their stable operating frequency and high quality factor. However, the disadvantages of quartz oscillators are their relatively large size and unsuitability for high integration with electronic circuitry (e.g., CMOS circuits).
Based on these limitations of conventional oscillators, there is a strong interest in the development of fully integrated silicon oscillators. Integration is important not only for reduced size but also reduced power consumption. It is possible to realize an integrated silicon oscillator using the mechanical properties of silicon devices. For example, silicon microelectromechanical (MEMS) resonators can provide small form factor, ease of integration with conventional semiconductor fabrication techniques and high f·C) products. High frequency and high-Q width-extensional mode silicon bulk acoustic resonators (SiBARs) and film bulk acoustic wave resonators (FBARs) have demonstrated atmospheric quality factors (Q) in excess of 10,000 at or above 100 MHz, with moderate motional resistances. Such resonators are disclosed in an article by S. Pourkamali et al., entitled “Low-Impedance VHF and UHF Capacitive Silicon Bulk Acoustic Wave Resonators—Part I: Concept and Fabrication,” IEEE Trans. On Electron Devices, Vol. 54, No. 8, pp. 2017-2023, August (2007), the disclosure of which is hereby incorporated herein by reference.
Unfortunately, such resonators may be characterized by relatively high temperature coefficient of frequency (TCF) values that require active compensation using temperature compensation circuits and/or relatively complex fabrication techniques to reduce TCF. Circuit-based compensation techniques typically increase the complexity of a resonator device and increase power consumption. Alternatively, fabrication-based compensation techniques that reduce TCF may cause a reduction in resonator quality factor (Q) and/or increase in resonator insertion loss. Examples of resonators that may use active and/or passive temperature compensation techniques are disclosed in U.S. Pat. Nos. 7,800,282, 7,843,284, 7,888,843, 7,924,119, 7,939,990, 7,955,885, 8,022,779, 8,061,013, 8,063,720 and 8,106,724, the disclosures of which are hereby incorporated herein by reference.
FIG. 2A illustrates a capacitive-type concave bulk acoustic resonator (CBAR) 20a, which is disclosed more fully in U.S. Pat. No. 8,063,720 to Ayazi et al, the disclosure of which is hereby incorporated herein by reference. The CBAR 20a includes a resonator body 22a (e.g., silicon) that is suspended opposite a recess (not shown) within a substrate 24a, 24b by a pair of opposing supports/tethers 15a, 15b located at opposite ends of the resonator body 22a. In contrast to the conventional bulk acoustic resonator 10 of FIG. 1, which includes a rectangular-shaped resonator body 12 suspended opposite a recess (not shown) in a substrate 14a, 14b, the CBAR 20a includes opposing concave-shaped sides. These sides curve inward relative to each other so that a minimum spacing at a center of the resonator body 22a is λ/2, where λ is a wavelength associated with a resonant frequency of the resonator body 22a. 
As further illustrated by FIG. 2A, a drive electrode (DRIVE) extends adjacent the first concave-shaped side of the resonator body 22a and a sense electrode (SENSE) extends adjacent a second concave-shaped side of the resonator body 22a. In contrast to the resonator 10 of FIG. 1, the width of the drive electrode (and sense electrode) as measured along the first concave-shaped side of the resonator body 22a is less than a spacing (5λ) between the first and second ends of the resonator body 22a. In particular, the width of the drive electrode as measured along the first concave-shaped side is preferably less than one-half a spacing between the first and second ends of the resonator body 22a and, more preferably, less than one-third a spacing between the first and second ends of the resonator body 22a. Moreover, when the widths of the first and second ends of the resonator body 22a are equal to 3λ/4, as illustrated by FIG. 2A, the opposing ends will not support acoustic energy at the resonant frequency determined by the central width λ/2. Accordingly, because of the concave shape of the sides extending adjacent the drive and sense electrodes, the resonator 20a operates to concentrate acoustic energy near the central width λ/2 of the resonator body 22a and thereby supports a high quality (Q) of the resonator 20a by reducing acoustic losses at the relatively narrow supports 15a, 15b. 
The concave-shaped resonator body 22a of the capacitive-type resonator 20a of FIG. 2A may be utilized within a piezoelectric-type resonator 20b of FIG. 2B. This resonator 20b is illustrated as including a piezoelectric layer 22b (e.g., ZnO) on a resonator body 22a (e.g., silicon) and first and second interdigitated electrodes 16a, 16b on the piezoelectric layer 22b. These electrodes 16a and 16b are patterned to include a plurality of finger-like extensions that operate to concentrate acoustic energy near the center of the resonator body 22a. As illustrated by FIG. 2C, the embodiments of FIGS. 2A and 2B may be combined to yield a concave bulk acoustic resonator (CBAR) 20c that supports both capacitive and piezoelectric transduction using electrostatic drive and sense electrodes and interdigitated electrodes 16a, 16b. 
A comparison of the TCF characteristics of the SiBAR of FIG. 1 versus the TCF characteristics of the CBAR of FIG. 2A is provided by FIG. 3. As illustrated by FIG. 3, a 100 MHz CBAR of FIG. 2A may exhibit a TCF of −6.31 ppm/° C., which is significantly lower than the TCF of −21.46 ppm/° C. of an otherwise equivalent 100 MHz SiBAR having a rectangular-shaped resonator body. These resonators were fabricated on the same boron-doped p-type silicon with a starting resistivity of about 0.001 Ω-cm. As illustrated by FIG. 4, the measured response of the CBAR in vacuum demonstrates a Q of 101,550 at 104.92 MHz and an fQ product of 1.06×1013.
One example of the CBAR 20c of FIG. 2C is more fully described in an article by Samarao et al., entitled ‘Combined Capacitive and Piezoelectric Transduction for High Performance Silicon Microresonators,’ MEMS 2011, Cancun, Mexico, January 23-27 (2011), pp. 169-172, the disclosure of which is hereby incorporated herein by reference. As illustrated by the graph of FIG. 5B herein, which is taken from FIG. 9a of the Samarao et al. article, a geometrically engineered CP-CBAR may be operated in: (1) a capacitive-drive and capacitive-sense mode using a 20 Volt polarization voltage (Vp) (e.g., by removing a piezoelectric stack from the concave resonator body); (2) a piezoelectric-drive and piezoelectric-sense mode; (3) a capacitive/piezoelectric-drive and capacitive/piezoelectric-sense mode; and (4) a capacitive-drive and piezoelectric-sense mode. As evident by this graph, the third mode (3) demonstrates highly efficient resonance at multiple frequencies, with reduced acoustic losses caused by spurious signal generation relative to mode (2). In particular, the third mode (3) produces two peak signals between 98 and 102 MHz, which have temperature coefficients of frequency that differ by 10 ppm/° C. The experimental configurations of these modes are more fully illustrated by FIGS. 3c and 6 of the Samarao et al. article and in a dissertation by Ashwin K. Samarao, entitled “Compensation and Trimming for Silicon Micromechanical Resonators and Resonator Arrays for Timing and Spectral Processing,” Dissertation Presented to the Academic Faculty, Georgia Institute of Technology, May 2011, published in depository archives on Jul. 6, 2011, pp. 1-132, the disclosures of which are hereby incorporated herein by reference.